# American Institute of Mathematical Sciences

April  2017, 14(2): 491-509. doi: 10.3934/mbe.2017030

## Modelling chemistry and biology after implantation of a drug-eluting stent. Part Ⅰ: Drug transport

 1 Mathematics Applications Consortium for Science and Industry, University of Limerick, Castletroy, Co. Limerick, Ireland 2 Department of Mathematics, University of Portsmouth, Winston Churchill Ave, Portsmouth PO1 2UP, United Kingdom 3 College of Engineering, Mathematics, and Physical Sciences, University of Exeter, Exeter, Devon, EX4 4QF, United Kingdom 4 School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, University Road, Galway, Ireland

Received  April 15, 2015 Accepted  July 26, 2016 Published  October 2016

Drug-eluting stents have been used widely to prevent restenosis of arteries following percutaneous balloon angioplasty. Mathematical modelling plays an important role in optimising the design of these stents to maximise their efficiency. When designing a drug-eluting stent system, we expect to have a sufficient amount of drug being released into the artery wall for a sufficient period to prevent restenosis. In this paper, a simple model is considered to provide an elementary description of drug release into artery tissue from an implanted stent. From the model, we identified a parameter regime to optimise the system when preparing the polymer coating. The model provides some useful order of magnitude estimates for the key quantities of interest. From the model, we can identify the time scales over which the drug traverses the artery wall and empties from the polymer coating, as well as obtain approximate formulae for the total amount of drug in the artery tissue and the fraction of drug that has released from the polymer. The model was evaluated by comparing to in-vivo experimental data and good agreement was found.

Citation: Tuoi Vo, William Lee, Adam Peddle, Martin Meere. Modelling chemistry and biology after implantation of a drug-eluting stent. Part Ⅰ: Drug transport. Mathematical Biosciences & Engineering, 2017, 14 (2) : 491-509. doi: 10.3934/mbe.2017030
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##### References:
The deployment of a drug-eluting stent in a diseased coronary artery. The stent is coated with a drug-loaded polymer, and subsequent to deployment of the stent, drug releases from the stent coating into the artery wall to prevent re-blockage due to restenosis
Drug diffuses from a polymer coating into the artery wall. In the arterial tissue, drug molecules can associate with and dissociate from specific binding sites, and can diffuse in their free form. Drug molecules may also be convected by the outward movement of plasma through the artery wall.
Numerical solutions to the initial boundary value problem (5-7) for various dimensional times $t=O(L_a^2/D_a)$, the diffusion time scale for free drug in the artery wall. The drug penetrates the artery wall on this time scale and this is evident in the figures. We have plotted profiles for the bound drug in the artery wall in (a), and the free drug in the polymer and the artery wall in (b). The parameter values used are $L=60, \varepsilon=10^{-6}$, $\eta = 0.002, K_b = 1700$, and $Pe=0.2$
Numerical solutions to the initial boundary value problem (5-7) for various dimensional times $t=O(L_p^2/D_p)$. On this time scale, the behaviour in the arterial tissue is quasistatic, and is driven temporally by the decreasing flux of drug from the stent coating. We have plotted profiles for the bound drug in the artery wall in (a), and the free drug in the polymer and the artery wall in (b). The parameter values used are $L=60, \varepsilon=10^{-6}$, $\eta = 0.002, K_b = 1700$, and $Pe=0.2$
Plots of the average bound drug, $m_a(t)$, in the artery wall as a function of time $t$ for various values of $\eta$, and with $L=60$, $\varepsilon=10^{-6}, K_b = 1700$, and $Pe=0.2$. On the vertical scale, $1$ corresponds to full occupancy of the specific binding sites. The rapid rise in the profiles near $t=0$ corresponds to the short time scale over which free drug in the tissue crosses the artery wall. In (a), $m_a(t)$ was calculated from numerical solutions of the initial boundary value problem (5-7). It is seen that there is significant occupancy of the binding sites for a period of a few months subsequent to the stent being implanted. In (b), we compare the asymptotic solution (28) to numerical results for the first month. The sub window shows the relative errors between the asymptotic and numerical solutions, $(m_{a,\text{numerical}}-m_{a,\text{asymptotic}})/m_{a,\text{numerical}}$
Comparison between in-vivo experimental data and modelling profiles for the fraction of the total drug released. In (a), Rapamycin release from Cypher [24] with $D_p=1.2 \times 10^{-10}$ mm$^2$/s and the mean squared error MSE $= 9 \times 10^{-4}$. In (b), Everolimus release from Xience V [20] with $D_p= 1.1 \times 10^{-11}$ mm$^2$/s and MSE $= 9 \times 10^{-3}$
Comparison of the average mass of drug in the artery wall calculated by (30) to in-vivo experimental data after (a) implantation of Cypher stents from 1 to 30 days [24] and (b) implantation of Xience V stents from 1 to 120 days [20]. Values have been normalized with respect to the initial drug content, $M(0)$. The parameter values used are: $L_a=0.45$ mm, $D_a=2.5 \times 10^{-4}$ mm$^2$/s, $V_a=5.8 \times 10^{-5}$ mm/s, $K_b = 300$; (a) $L_p=1.26 \times 10^{-2}$ mm, $M(0)= 174.89$ $\mu$g, $D_p=6.0 \times 10^{-11}$ mm$^2$/s, $\eta = 0.0075$ (MSE=$1.7 \times 10^{-4}$); and (b) $L_p= 7.6 \times 10^{-3}$ mm, $M(0)= 100$ $\mu$g, $D_p=1.0 \times 10^{-11}$ mm$^2$/s, $\eta = 0.002$ (MSE=$2.2 \times 10^{-4}$)
Data for some commercial drug-eluting stents
 DES/Drug Polymer thickness ($L_p$) Drug dose Life time References Cypher/Rapamycin 12.6μm 140 μg/cm2 stent surface area 80% of drug released within 30 days [7,14] Taxus/Paclitaxel 16 μm 100 μg/cm2 stent surface area Early 48 hours burst, then slow release over 10 days [7,14] Endeavor/Zotarolimus 5.3 μm 100 μg/cm stent length 95% of drug released within 15 days [7,14 Xience V/Everolimus 7.6 μm 100 μg/cm2 stent surface area 80% of drug released within 30 days [7,14] Promus Element/Everolimus 7 μm 100 μg/cm2 stent surface area 80% of drug released within 30 days [3]
 DES/Drug Polymer thickness ($L_p$) Drug dose Life time References Cypher/Rapamycin 12.6μm 140 μg/cm2 stent surface area 80% of drug released within 30 days [7,14] Taxus/Paclitaxel 16 μm 100 μg/cm2 stent surface area Early 48 hours burst, then slow release over 10 days [7,14] Endeavor/Zotarolimus 5.3 μm 100 μg/cm stent length 95% of drug released within 15 days [7,14 Xience V/Everolimus 7.6 μm 100 μg/cm2 stent surface area 80% of drug released within 30 days [7,14] Promus Element/Everolimus 7 μm 100 μg/cm2 stent surface area 80% of drug released within 30 days [3]
Drug diffusivities, $D_a$, in arterial tissues
 Drug Diffusivity $D_a$ (mm2/s) References Rapamycin $1.5 - 2.5 \times 10^{-4}$ [24] Paclitaxel $2.6 \times 10^{-6}$ [28] Dextran $3.0 \times 10^{-5}$ [12] Heparin $7.7 \times 10^{-6}$ [13]
 Drug Diffusivity $D_a$ (mm2/s) References Rapamycin $1.5 - 2.5 \times 10^{-4}$ [24] Paclitaxel $2.6 \times 10^{-6}$ [28] Dextran $3.0 \times 10^{-5}$ [12] Heparin $7.7 \times 10^{-6}$ [13]
Data for drug diffusivities in polymers. Note: PEVA = Poly(ethylene-co-vinyl acetate), PBMA = Poly(n-butyl methacrylate), PVDF-HFP = Poly(vinylidene fluoride-co-hexafluoropropylene), SIBS = Poly(styrene-b-isobutylene-b-styrene)
 Drug Diffusivity $D_p$ (mm2/s) Polymer DES References Rapamycin $1.2 \times 10^{-10}$ PEVA and PBMA Cypher This study $6.3 \times 10^{-11}$ PEVA and PBMA Cypher [16] Everolimus $1.0 - 1.2 \times 10^{-11}$ PBMA and PVDF-HFP Xience V This study Paclitaxel $O(10^{-15}) - O(10^{-11})$ SIBS Taxus [22]
 Drug Diffusivity $D_p$ (mm2/s) Polymer DES References Rapamycin $1.2 \times 10^{-10}$ PEVA and PBMA Cypher This study $6.3 \times 10^{-11}$ PEVA and PBMA Cypher [16] Everolimus $1.0 - 1.2 \times 10^{-11}$ PBMA and PVDF-HFP Xience V This study Paclitaxel $O(10^{-15}) - O(10^{-11})$ SIBS Taxus [22]
Data for transmural velocities and pressures in the arterial wall
 Artery Transmural velocity ($\times 10^{-5}$ mm/s) Transmural pressure (mmHg) References Porcine coronary 5.8 50 [24] Rabbit carotid 1.85±0.33 110 [11] 8.9±6.8 60 [1] Rabbit thoracic aorta 2.8±0.9 70 [23] 4.4±1.4 180 [23] Rabbit femoral artery 3.3±1.3 30 [2] 8.1±2.4 60 [2] 9.9±2.5 90 [2]
 Artery Transmural velocity ($\times 10^{-5}$ mm/s) Transmural pressure (mmHg) References Porcine coronary 5.8 50 [24] Rabbit carotid 1.85±0.33 110 [11] 8.9±6.8 60 [1] Rabbit thoracic aorta 2.8±0.9 70 [23] 4.4±1.4 180 [23] Rabbit femoral artery 3.3±1.3 30 [2] 8.1±2.4 60 [2] 9.9±2.5 90 [2]
Values for some of the non-dimensional parameters appearing in the model for some commercial drug/stent systems. For the purposes of calculation, the values $L_a=0.75$ mm and $V_a=6 \times 10^{-5}$ mm/s [24] have been chosen. For Rapamycin, $k_{\rm {off}} = 0.096$ min$^{-1}$, $k_{\rm {on}} = 4.8 \times 10^{7}$ M$^{-1}$ min$^{-1}$, and $b^*= 3.3 \times 10^{-6}$ M [27,24]. For Paclitaxel, $k_{\rm {off}} = 5.46$ min$^{-1}$, $k_{\rm {on}} = 2.2 \times 10^{8}$ M$^{-1}$ min$^{-1}$, and $b^*= 1.0 \times 10^{-5}$ M [27,28]. The remaining values used can be found in Tables 1 and 2
 Stent/Drug $L=L_a/L_p$ $Pe=V_aL_a/D_a$ $K_b=k_{\rm {on}} b^*/k_{\rm {off}}$ Cypher/Rapamycin 60 0.2 1700 Taxus/Paclitaxel 47 17 400 Endeavor/Zotarolimus 141 0.2 1700 Xience V/Everolimus 99 0.2 1700 Promus Element/Everolimus 107 0.2 1700
 Stent/Drug $L=L_a/L_p$ $Pe=V_aL_a/D_a$ $K_b=k_{\rm {on}} b^*/k_{\rm {off}}$ Cypher/Rapamycin 60 0.2 1700 Taxus/Paclitaxel 47 17 400 Endeavor/Zotarolimus 141 0.2 1700 Xience V/Everolimus 99 0.2 1700 Promus Element/Everolimus 107 0.2 1700
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